An accurate elementary mathematical library for the IEEE floating point standard

Abstract

The algorithms used by the IBM Israel Scientific Center for the elementary mathematical library using the IEEE standard for binary floating point arithmetic are described. The algorithms are based on the “accurate tables method.” This methodology achieves high performance and produces very accurate results. It overcomes one of the main problems encountered in elementary mathematical functions computations: achieving last bit accuracy. The results obtained are correctly rounded for almost all arguement values. Our main idea in the accurate tables method is to use “nonstandard tables,” which are different from the natural tables of equally spaced points in which the rounding error prevents obtaining last bit accuracy. In order to achieve a small error we use the following idea: Perturb the original, equally spaced, points in such a way that the table value (or tables values in case we need several tables) will be very close to numbers which can be exactly represented by the computer (much closer than the usual double percision representation). Thus we were able to control the error introduced by the computer representation of real numbers and extended the accuracy without actually using extended precision arithmetic.